Integrals of Vector-valued Functions

 Goal: Calculate the integral of vector-valued functions

Integrals of vector-valued functions

The antiderivative and definite integral of vector-valued functions can be calculated by taking the antiderivative and definite integral of the real-valued functions. This is the same way we have been doing with the derivative.

The antiderivative of a vector-valued function appears in applications. For example, if a vector-valued function represents the velocity of an object at a time t, then its antiderivative represents its position. Or, if the function represents the acceleration of the object at a given time, then the antiderivative represents its velocity.

Definition

Let f, g, h be real-valued functions integrable over the closed interval [a, b].

1. The indefinite integral of a vector-valued function r(t) = f(t)i + g(t)j is given by:

∫[f(t)i + g(t)j]dt = [∫f(t)dt]i + [∫g(t)]j

The definite integral of a vector-valued function is:

2. The indefinite integral of a vector-valued function r(t) = f(t)i + g(t)j + h(t)k is given by:

∫[f(t)i + g(t)j + h(t)j] = [∫f(t)]i + [∫g(t)]j + [∫h(t)]k

The definite integral of a vector-valued function is:

Examples

Calculate each of the following integrals:

Solution

a. Let's use the first part of the definition of the integral of a space curve:

Practice

Calculate the following integral: